Rental car crossing into France but Return to original spot in Barcelona

Taking it across the border is not costly, there is usually no charge. But check with the rental company when you book, just in case.
What is costly is returning it in a different country to where you picked it up.

I looked up Galinagues, is in Aude, near the Spanish border and Andorra. Were you planning to drive to Paris?
I suggest you drive back to Barcelona, turn the car in there and get either a TGV (High Speed Train) or fly to Paris.
Alternative, if you want to drive to Paris stopping off on the way, is from Barcelona, get a train to (e.g.) Perpignan, pick up the car there and return it when you arrive in Paris.
No way drive in Paris (or in Barcelona).

Check with the car rental company. I have taken a car from Portugal into Spain on several occasions. Sometimes I have had to pay a small additional daily fee for the time it isn’t in Portugal and with other providers, it’s included in the base rental price.

We picked up our car in Vigo in Galicia, Spain and dropped it off in Barcelona. During the trip we explored the French Basque area, drove along the French side of the Pyrenees, visited Carcasonne and Collioure before re entering Spain. There are no border crossings, no issues with anything. No worries.
The car should be returned to the country where it was rented or a huge drop fee will be charged. We were told €1000 fee. So, they need to return the car to Spain. They also can take a train or taxi into France and pick up a car there if not returning to Spain.
(This trip actually began in Lisbon and we dropped the Portugese rental off in Porto, taxi into Spain, picked car up in Vigo.)
Paris- I would not want to stay nor my daughter to stay in Monmarte. Look at the Marais, and the 5th or 6th arrondisement.

I’ve never had an issue driving across country borders. It would be costly if you rented in Spain and returned in France, but not returning to the same location. Years ago I drove in Paris and had to drive through that traffic circle at the Arc de Triomphe twice. Made it through but it was nerve wracking. On that trip I took the car from the Netherlands to France, Belgium, Luxembourg, and Germany, using the Netherlands as a base. No problems and no extra charge.

In general you can drive it across European borders freely, but do check to be sure. Renting a car in Salzburg I was notified that there were fees to take it into Slovakia and maybe other Eastern Europe countries but I could take it between Austria, Germany etc as much as I wanted. I don’t know exactly which countries were allowed because I was just using it in Austria. I am pretty sure Spain and France would be no issue, but as I said, confirm.

I wouldn’t worry about Barcelona traffic for getting/returning a car. You don’t want to use a car to get around, but getting in and out of the city to a rental agency isn’t much of an issue if you are used to city driving.

I've driven cross country borders within Europe many times all without any additional expense or problems but have always returned the car to the original rental place.

I haven't driven in Barcelona but I'd have no issue in doing so, it's nowhere near as bad as Rome for example.